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Diffusion-Scale Tightness of Invariant Distributions of a Large-Scale Flexible Service System

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  04 January 2016

A. L. Stolyar
A. L. Stolyar*
Affiliation:
Bell Laboratories
*
∗ Current address: Lehigh University, Mohler Laboratory, 200 West Packer Avenue, Bethlehem, PA 18015, USA. Email address: [email protected]
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Abstract

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A large-scale service system with multiple customer classes and multiple server pools is considered, with the mean service time depending both on the customer class and server pool. The allowed activities (routeing choices) form a tree (in the graph with vertices being both customer classes and server pools). We study the behavior of the system under a leaf activity priority (LAP) policy, introduced by Stolyar and Yudovina (2012). An asymptotic regime is considered, where the arrival rate of customers and number of servers in each pool tend to ∞ in proportion to a scaling parameter r, while the overall system load remains strictly subcritical. We prove tightness of diffusion-scaled (centered at the equilibrium point and scaled down by r−1/2) invariant distributions. As a consequence, we obtain a limit interchange result: the limit of diffusion-scaled invariant distributions is equal to the invariant distribution of the limiting diffusion process.

Keywords

Many-server modelpriority disciplinefluid limitdiffusion limittightness of invariant distributionlimit interchange

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60F17: Functional limit theorems; invariance principles
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 1 , March 2015 , pp. 251 - 269
Copyright
© Applied Probability Trust 

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